Show that f(x)=e^x from set of reals to set of reals is not invertible…

Yes, this is my question...

How can you prove this? That $f(x)=e^x$ from the set of reals to the set of reals is not invertible, but if the codomain is restricted to the set of positive real numbers, the resulting function is invertible. As far as I know, this function is one to one over its entire domain...which means it is invertible.

Invertible means one-to-one and onto. In particular, we only say that a map $f:A \to B$ is invertible if there is another map $g:B \to A$ such that both $f \circ g$ and $g \circ f$ are the identity maps over their respective spaces.
the function is not surjective as no $-ve$ number has a preimage hence the conclusion